A New Interface Tracking Method: Using Level Set and Particle Methods Advisors: Jinsun Sohn, Joseph Teran (UCLA) Trevor Caldwell 2 Noah Duncan 2 Tengyuan Liang 3 Shirley Zheng 1 1 Cornell University 2 Harvey Mudd College 3 Peking University August 3, 2011 Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 1 / 34
Table of Contents I Introduction 1 Problem Statement Frames of Reference Particle Methods 2 Existing Methods 3 Particle Level Set Grid-Based Particle Variational Method Our Method: Dynamic Reconstruction Method 4 Procedure Performance Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 2 / 34
Background Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 3 / 34
Background Typical traditional interface advection is done using level set or particle methods. Because of numerical error, objects lose volume when advected by level set methods so that they shrink and eventually disappear as time passes. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 4 / 34
Background In contrast, particle methods are numerically accurate but do not naturally handle topological change. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 5 / 34
Problem Statement Devise a method that is numerically accurate and robust under topological change . The interface should be represented implicitly at each iteration, and the particles might be resampled periodically. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 6 / 34
Implicit vs Explicit Representation We can capture and evolve a hypersurface implicitly using level set methods, or we can track it explicitly using particle methods. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 7 / 34
Implicit vs Explicit Representation We can capture and evolve a hypersurface implicitly using level set methods, or we can track it explicitly using particle methods. In level set methods, we update the implicit function φ across the domain at each time step. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 7 / 34
Level Set Figure: Signed Distance Representation of Circle in R 2 . Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 8 / 34
Implicit vs Explicit Representation We can capture and evolve a hypersurface implicitly using level set methods, or we can track it explicitly using particle methods. In level set methods, we update the implicit function φ across the domain at each time step. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 9 / 34
Implicit vs Explicit Representation We can capture and evolve a hypersurface implicitly using level set methods, or we can track it explicitly using particle methods. In level set methods, we update the implicit function φ across the domain at each time step. In particle methods, we update the location of the zero-isocontour at each time step. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 9 / 34
Particles Figure: Explicit Representation of Circle in R 2 Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 10 / 34
Level Set and Particle Method: Tradeoffs Level Set Method Strengths Maintains distance function Handles topological changes Weaknesses Computationally expensive Numerically Inaccurate Particle Method Strengths Computationally efficient and numerically accurate Weaknesses Does not handle topological change (e.g. merging/pinching) Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 11 / 34
Existing Methods Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 12 / 34
Existing Methods Interface Evolution Schemes Particle Level Set Method (Enright, Fedkiw, Ferziger, and Mitchell) Grid-Based Particle Method (Leung and Zhao) Surface Reconstruction Schemes Fast Variational-Based Surface Reconstruction (Ye, Bresson, Goldstein, and Osher) Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 13 / 34
Particle Level Set Method Seeds particles in a positive and negative band around the level set. Detects errors in level set when a particle crosses into a region of opposite sign. Overwrites level set signed distance function with particle signed distance function. Superior to basic level set method in preserving volume. Difficult to implement and reseeding strategy is not robust. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 14 / 34
Particle Level Set Method PLS vs Level Set Method. Light red and light blue particles are escaped particles. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 15 / 34
Grid-Based Particle Method Uses Eulerian information from grid cells to compute curvature and normal vectors for particles. Uses a least-squares quadratic fit to approximate the interface locally. Lacks a robust method for determing inside / outside information for interface, but accurately computes distance. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 16 / 34
Variational Method Estimates a surface from a set of unorganized scattered points using variational methods Solves an inverse edge detection problem to obtain an initial surface estimate We adapt this step of the method to align a level set with particles Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 17 / 34
Our Method: Dynamic Reconstruction Method Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 18 / 34
Dynamic Reconstruction Method Algorithm 5.1: DynamicReconstruction ( T , k , φ ) for t = 1 : T if ( t mod k ) == 1 (re)seed particles on zero-isocontour of φ ; end advect particles; calculate distance function φ using the particles; compute sign (inside/outside) information using an edge detector; end Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 19 / 34
1. (Re)seed particles on the zero-isocontour GREEN dots mark exterior grid points, RED dots mark interior grid points. Unless a grid cell has unanimous sign on all grid points, linearly interpolate the zero-crossing on each of the 6 edges. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 20 / 34
2. Advect Particles Advect particles using 2 nd order Runge-Kutta scheme: for t = 1:T ∀ p ∈ { Particles } do: x 0 = x ( p ); y 0 = y ( p ); x 1 = x 0 + v x · dt ; y 1 = y 0 + v y · dt ; x 2 = x 1 + v x · dt ; y 2 = y 1 + v y · dt ; x ( p ) = x 2 + x 0 ; y ( p ) = y 2 + y 0 ; 2 2 end Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 21 / 34
3. Compute Distance Using Particles Fast-Sweeping Algorithm (Zhao, 2004) We want to reconstruct the zero-isocontour using the particles. 1. Initialization. On cut grid cells: calculate distance to the interface using the particles. On the rest of the domain: initialize to a large constant, c . Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 22 / 34
3. Compute Distance Using Particles Fast-Sweeping Algorithm (Zhao, 2004) We want to reconstruct the zero-isocontour using the particles. 1. Initialization. On cut grid cells: calculate distance to the interface using the particles. On the rest of the domain: initialize to a large constant, c . 2. Discretize, and iteratively sweep the domain in alternating directions. At each interior grid point, solve the eikonal equation x min ) + ] 2 + [( d h [( d h i , j − d h i , j − d h y min ) + ] 2 |∇ d | 2 ≈ = f 2 i , j h 2 and update ¯ d i , j to be the minimum between c and the computed solution. Use one-sided difference on the boundary. Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 22 / 34
3. Compute Distance Using Particles We solve |∇ d | = 1 using the fast sweeping algorithm: Ready Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 23 / 34
4. Calculate Sign Information Using an Edge Detector Surface Reconstruction Using the Eikonal Equation and the Chan-Vese Model (Ye, Bresson, Goldstein, Osher, 2010) We want to approximate a two-valued function f whose edges are located along the set of particles, that is � for x ∈ Ω + , < C f ( x ) ≥ C for x ∈ Ω − . where 0 . 5 < C < 1 is a critical value that segments the domain. Let d ( · ) be the unsigned distance function. Observe that −∇ d is equivalent to a vector flow pointing toward the interface computed from particles. d ( · ) is an edge detector function . Using ǫ = d x p for stability, we can approximate the image f to this edge detector by solving the eikonal equation: 1 |∇ f | = for p = 3 d p + ǫ Caldwell, Duncan, Liang, Zheng (UCLA REU) Computational Interface Group August 3, 2011 24 / 34
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